Animated textures don't work in Blender viewport playback
It is a very, very, very, very well known limitation of the current system.

Viewport shaders are optimized for speed and performance for smooth playback and as responsive interaction as possible during animations.

Animated materials and textures require shader re-compilation, and this is a computationally expensive and resource heavy task, that is generally preformed on the CPU rather than the graphics card). Recompiling requires constant re-upload and sending data back and forth from the CPU to the GPU which is comparatively slow.

As such, they are pre-compiled before playback to avoid constant updates or refreshes, and during this process animations are suppressed to achieve faster updates and increased smoothness during playback.

It is a limitation that will likely be overcome in future 2.8 versions of Blender and the redesigned EEVEE viewport engine.

That being said, if you plan on rendering using the OpenGL renderer whether for your final output or animation previews, offline rendering will effectively update the animated materials and correctly render your movie.

If you only have a sequence of images or a movie strip as texture just turn on the option Auto Update in the image texture node